Windows and Mirror Symmetry

多项式方程的一组解形成几何形状（如直线、平面、曲线）。代数几何研究这些多项式方程的解与这些形状的几何之间的联系。辛几何也研究几何形状，但不考虑它们可能具有的任何代数性质，而是保留一种测量内部面积的方法。在20世纪90年代早期，高能理论物理学家发现辛几何中的问题可以用代数几何来回答，反之亦然。这两个数学领域之间的联系被称为“镜像对称”。1994年，Maxim Kontsevich在他的菲尔兹奖演讲中提出了一个庞大的镜像对称数学框架，通过范畴论（一种可以用来描述大多数数学主题的现代数学语言）将代数几何和辛几何联系起来。康采维奇的镜像对称公式是完全革命性的，在数学上没有任何先例。在过去的25年里，它已经发展成为一个庞大的研究领域，吸引了来自代数几何、辛几何和高能理论物理等不同国际领域的研究人员。作为他的计划的重要组成部分，Kontsevich将镜像对称的代数几何方面提炼成一个被称为派生范畴的数学概念。本研究计划旨在提供衍生类别的基本描述，特别是当它们出现在高能理论物理模型中时。它将衍生的类别分解成更简单的部分，在它们之间进行比较，并旨在证明有关它们行为的一些最大问题。它还在Kontsevich的框架内构建了镜像对称的新版本。最后，将所得理论应用于求解代数几何中的经典问题。

The set of solutions to polynomial equations form geometric shapes (e.g. lines, planes, curves). Algebraic geometry studies the connection between the solutions to these polynomial equations and the geometry of these shapes. Symplectic geometry also studies geometric shapes but disregards any algebraic natural they may have, retaining instead a way to measure the area inside. In the early 1990s, high-energy theoretical physicists found that questions in symplectic geometry can be answered using algebraic geometry and vice versa. This connection between these two areas of mathematics is called "mirror symmetry". In 1994, during his Fields Medal address, Maxim Kontsevich proposed a vast mathematical framework for mirror symmetry, connecting algebraic geometry and symplectic geometry through category theory, a modern mathematical language which can be used to describe most mathematical topics. Kontsevich's formulation of mirror symmetry was completely revolutionary and without any precedent in mathematics. In the last 25 years, it has burgeoned into a massive field of study attracting a diverse international field of researchers from algebraic geometry, symplectic geometry, and high-energy theoretical physics. As an essential part of his program, Kontsevich distilled the algebro-geometric side of mirror symmetry into a mathematical concept known as the derived category. This research proposal aims to provide a fundamental description of derived categories, especially as they appear in models from high-energy theoretical physics. It decomposes derived categories into simpler pieces, makes comparisons between them, and aims to prove some of the biggest questions remaining about their behavior. It also constructs new versions of mirror symmetry within Kontsevich's framework. Finally, it aims to apply the resulting theory to solve classical problems in algebraic geometry.